Local and global well-posedness results for the Cauchy problem
(REF ) have been extensively analysed in Sobolev spaces
(see for instance
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}),
where the optimal results in \(H^{1/2}(\mathbb {R})\)  were deduced by
H. Takaoka [10]} and
H. Bahouri and G. Perelman [1]}, respectively.
